Space of holomorphic functions of polynomial growth as local algebra

Let G be a domain in the complex plane, star-shaped with respect to the point 0, H-∞(G) be the space of holomorphic functions in G of polynomial growth near the boundary of G. The Duhamel product ∗ is introduced in it. This product is used in operational and operator calculus, in the spectral theory, in the problem of the spectral multiplicity of a linear operator, in boundary value problems. It is shown that H-∞(G) with it is a unital topological algebra. The integration operator J(f)(z)=∫z0f(t)dt acts linearly and continuously in H-∞(G). It is proved that all linear continuous operators in H-∞(G) that commute with J, are represented as Sg(f)=f∗g, where g is a fixed function from H-∞(G). In the case where G is strictly star-shaped with respect to zero, a criterion for the invertibility of an element of the algebra H-∞(G) and a criterion for the operator Sg to have the continuous linear inverse are proved. It is shown that every nonzero operator from the commutator subgroup J is a composition of the power of the operator J and some isomorphism from the aforementioned commutator subgroup. In the proving of ∗-invertibility the Neumann series is used, usually applied in Banach spaces. In non-normable locally convex spaces of functions it was previously used by L. Berg, N. Wigley, and M. T. Karaev. All closed ideals of the algebra (H-∞(G),∗), closed invariant subspaces and cyclic vectors of J in H-∞(G) are described. From the obtained results it follows that the operator J is unicellular and the algebra (H-∞(G),∗) is local. The only maximal ideal in it is the set of all ∗-irreversible elements.